July 2010

The book begins by giving a very quick overview of derivatives pricing, stochastic vol models, risk neutral pricing and ends the first chapter by giving couple of motivating examples where loosely coupled joined distributions given marginals , are used to price options .

Chapter 2: Bivariate Copulas

Copula is a type of function . But a very special function which maps marginals to joint. Hence one needs to be aware about the nature of the function and also its limitations. In that sense, this chapter develops copula function from scratch by explaining in detail the various conditions for a function to be called a copula. Boundary conditions of a generic copula are introduced using Frechet bounds. The good thing about this book is that it tries to give a real life application at the right stages of the book so that a reader can immediately relate the concept. For example the Frechet bounds are technically introduced in most of the books(Nelson). However the authors in this book cite an example of an exotic option where Frechet bounds are used to calculate the minimum and maximum value of the option. Various types of copulae are introduced such as subcopula, minimum copula, maximum copula, product copula, Survival copula, Joint Survival copula for uniform variates etc. The highlight of this chapter is explain Sklar’s theorem in a detailed manner.

Absolutely continous condition and Singularity conditions are discussed in the context of a bivariate copula . Also mentioned is the canonical representation which links the density of the joint distribution, density of marginals and the copula.

Various examples showing the application of copula to option pricing / credit risk(this area has made copula notorious, thanks to the article in wired titled –“The Formula that killed the Wall Street”.These examples would make any reader curious about the inner math of copula

Chapter 3 : Market Comovements

This chapter is extremely important as it introduced nonparametric measures of dependence like Kendall’s tau and Spearman Rho’s in the context of Copulas. One will develop a healthy dose of scepticism towards the simple correlation estimates that one sees in our daily lives often. One can clearly see that linear correlation that most of us know , is so narrowly defined .The book will make the reader pause and think of the ton of assumptions behind the all pervasive metric.

The second part of this chapter introduces various forms of copulae such as gaussian, student t, frechet family, Archimedean family by giving the connections between the marginals and the joint.

Chapter 4: Multivariate Copulas

This chapter extends bivariate copulas to multi dimensional case. Basically all the theorems used in the bivariate case are written in multivariate world.

One important aspect mentioned in this chapter is Density and Canonical representation of a multidimensional copula. This is extremely important when one needs to find out conditional distributions of copulas for generating random numbers for various copulae.

Chapter 5: Estimation and Calibration from Market Data

One of the challenging aspects of estimating a copula model from a set of securities is the sheer number of parameters that needs to be estimated and the way to compare fitness statistics across copulas. Saviour is good old MLE. However MLE also becomes unstable if an estimation is made all at once. Hence the estimation is generally split in two steps.

First Step involves estimating parameters for the assumed marginals, which can be assumed as normal/ t / gamma / GARCH etc . Second Step involves estimating the copula function given these parameters for the marginals. This procedure is called Inference functions for the Margins(IFM).

The second method described is a bit more non parametric. The method is called CML , i.e Canonical Maximum Likelihood. The key step in this process is to estimate the empirical distribution function for the margins and use it in the second step of estimating copula parameters. Where can one use this estimation procedure ? What if you don’t want copula in your strategies/ algos / modeling ? In that case, copula can serve as a superb diagnostic tool of your assumptions. For example lets say you are interesting in creating a portfolio of a few stocks from a specific sector. So, you estimate basic moments of each of the assets and then make a ballpark estimate of risk using correlation matrix. Can you trust this correlation matrix ? Well one way to answer this question is let’s say you fit a Student t marginal to the stocks in your portfolio and fit a gaussian copula. The dependence structure of the fitted copula will give tons of insight in to the assumed correlation. Should you use robust estimators ? Should you stick to the age old product moment correlation matrix ? All these questions can be answered by fitting various marginals to the gaussian copula. Agreed we are in the end fitting a gaussian copula which has no tail dependence, but atleast you are introducing symmetric tail dependence in the individual securities. Another application is fitting a stochastic process to the individual securities and then joining them with an appropriate copula. This kind of loose coupling of marginals and joints give a modeller tremendous amount of flexibility. One can also think of a time varying copula function.

Chapter 6 :Simulation of Market Scenarios

Typically in any market of n securities, each security follows a stochastic path. The most generic form of modeling all the variables together is to Fit N different marginals to the n different securities and then connect the N marginals with a time varying Copula function. Each of these marginals can in turn be a conditional marginal which will complicate the things. So, any quant exercise which involves data analysis/modeling/simulation with more than one security obviously needs to have this general framework in mind and one has to clearly state the assumptions that are made.

Surprisingly this generic framework is not clearly stated in most of books and one typically finds a situation where margins are normal and copula is guassian normal copula, meaning the joint distribution is multivariate normal. Whenever one does modeling one should always keep in mind that the Gaussian framework is a strawman , which is to be taken as example101 for the generic framework.

For any portfolio strategy involving multiple assets, backtesting a strategy necessitates simulation of alternate worlds. While sampling with replacement is a wonderful way to test a strategy, at the same time, it is sometimes very harsh and one ends up simulating multivariate normal which I think is too lenient. The former might give rise to Type II error while latter can give rise to Type I error. In that context, this chapter is extremely useful to simulate various market scenarios. The usage of copulas gives the flexibility of modeling tail dependence. Methods of generating random variables for various copula like Gaussian copula, Archimedean Copulae such as Gumbel, Clayton and Frank are described in great detail.

For generating an n variate Gaussian copula, the procedure is similar to generating a multivariate random variable with a given mean vector and covariance vector. With a few additional steps to the multivariate normal sample, one can easily generate a Gaussian copula.

For generating an n variate Student T copula, the procedure is again straightforward , starting with generating a multivariate random number and then dividing by average chi-square random number.However generating n dim random numbers for any of the Archimedean Copula is not straightforward. Iterative procedures on conditional distributions are used. In some cases like Clayton and Frank Copula, there are no closed forms that form the part of iterative procedure. However for Gumbel there is no closed form solution for evaluating the various random numbers in the iterative process.

The last 2 chapters cover a range of applications in the credit world and exotic option pricing world.

Takeaway

Any modeller deals with more than one variable in his work. This book will help one see the powerful use of joint distributions. Also, It is definitely the most accessible introduction to Copulas which will help a reader be more comfortable with reading and understanding the applications of Copulas in various fields.

Just because Guassian Copula was used in pricing CDS and CDO and they all went bust, doesn’t mean Copulas as a concept is flawed. Well, what is flawed is substituting a million correlations amongst mortgages that needs to be fed in to the model , with ONE Single correlation . That is a plain misapplication of the concept.

My takeaway is that Copulas are excellent diagnostic tools for analyzing joint behaviour of variables and they must be a part of Quant’s arsenal.

Bounce is written by Matthew Syed , a columnist for the Times of London and a BBC Commentator. More importantly, he is a three time commonwealth table tennis champion , a two-time Olympian. Stuff from a sportsman is something one can always read as it is a story told by someone who actually was in the trenches and can tell a better story I think , than a mere spectator. Its debatable point though whether narrative is better from a sportsman or a bystander. I tend to think the former is better.Anyways, coming back to this book, it has three parts.

Talent Myth :

Mathew Syed starts off the book with “iceberg-illusion” , where we see an outstanding display of talent in a person and appreciate the end product as innate talent but fail to appreciate that one is witnessing an end product of a process measured in years. His main argument like the previous literature is that , QUANTITY of practice matters. He cites examples from the lives of Federer, Agassi, Polgar sisters, Williams Sisters, Tiger Woods, etc to drive home the point that it is quantity and quality of practice that separates really successful players.

I liked a few examples from a chapter titled – “Power and Impotence of Practice”. Practice with out involvement is actually impotent, says the author. Whenever you do a task in an auto-pilot mode, there is nothing to be learnt in the task. Only when we try to consciously come out of auto pilot mode and deep practice the task, one can gain mastery over stuff. There are so many people who would have spent hours and hours doing a specific task but the sad thing i guess is, it makes them move to auto-pilot mode quickly and they do not get out of that mode from time to time to master the skill.

Another example from this book which was appealing to me, was an anagram puzzle. It goes something like this.

Here is a list of anagrams , crack them quickly

FAHTER

FOOTBLAL

DCOTOR

OUTCOEM

TEACHRE

Now Here is another list of anagrams, crack them quickly

HERFAT

LBOFTOAL

RTOCOD

ECMUTOO

EERTACH

If you actually solved anagrams from both the lists, you will have notices that they actually refer to precisely the same words. Here is the curious thing : When researchers had participants work on first list and later questioned, they were not good at remembering words. However the participants who worked on second list had a good recall rate .WHY ? In the second list, you are forced out of auto pilot and hence the word is imprinted on your memory.

Simple example but proves the point that , it is better to solve hard and challenging problems than easy ones. Hence practice in whatever field it might be should not be effortless.. If it is so , then there is a great chance of getting in to auto-pilot mode

The first part of the book ends with citing the research of Carol Dweck of Stanford University who champions for the cultivation of growth mind set than fixed mindset.

Overall, the first part of the book can be called as a nice commentary on three books that have already been published on talent myth.

Paradoxes of the Mind :

The second part of the book starts with Placebo effect. Syed gives several examples of athletes around the world who completely believe in themselves before taking a shot, irrespective of the level of the competition or the competency/ranking of the opponent. This is termed as Placebo effect in the book as it helps the player play well and have a positive attitude towards the game. “Double think”, the act of deliberating before making a choice in a game, and subsequently having complete belief in oneself that he/she is going to make it, are two opposing thoughts that outstanding players regularly have. In fact, the capacity to have this kind of a mindset, is probably cultivated through thousands of hours of deep practice.

Syed talks about choking, times when highly accomplished successful players play like novices. As per his research, this typically happens when a player gives extreme importance to the event. It results from too much focus. One of the remedies for choking is to have a mindset – “outcome doesn’t matter”. Easier said/written than done. Basically it means, “Playing as if it means nothing when it means everything”

The author ends the second part of the book by talking about superstitions that some of the successful players believe in. Well, as long as they help them in maintaining their calm and having a clear mindset for the game, there is no harm in it I guess. Then , there is the aspect of ,”What drives a player once they reach No 1 ?” For many , it is a void and then they quickly lose their No 1 spot. For only a precious view, it is something that they were not expecting .The joy of playing a game & improving their game is far more important than medals and ranking.

Deep Reflections :

The third part of the book starts off with discussing extraordinary sense of perception amongst top players and says that it is the quantity and quality of practice that allows extra bandwidth of attention. Top players thus use this bandwidth to perceive faster, smarter and deeper. The argument is the same as made in a book title Flow by Mihaly Csikszentmihalyi. As one keeps doing tasks over long periods, one can perform certain kind of tasks in auto-pilot mode and hence can free up their mind space to perceive other matters of the game.

There is a random chapter on the use of Drugs , ethical issues etc , which I felt was unnecessary in the book as it did not gel with the flow of the book.

The last chapter of the book tries to answer the most popular question , “Are Blacks Superior Runners ?”. As easy it is to give a race based answer, the author cites interesting sources which goes to show that , it is not that the entire East African nation has good distance runners or West African nation has good sprinters. There are few talent hot beds which produce amazing players year after year. Analysis of these hot beds reveals that it has nothing to do with genes but very specific talent hot bed characteristics. For example, one small district Nandi, with only 1.8% of Kenya’s population produced half of world class distance running athletes. Why ? reason, is similar to Gladwell a.ka. Outlier reasoning… Kenya’s top runners have to run an excess of 20 km per day to attend to school because there is no other means to reaching school and that too , running on an altitude. Hence by the time they reach their 16th birthday, each would have clocked 6000 hours of running that too altitude running. This as per a few analysis and author’s argument is the reason for the success of Kenyan runners.

Here are some audio and video links for those who are time strapped and don’t have the time to read the book.

The first part of the book is mashup of “Outliers”, “The Talent Code” and “Talent is Overrated”, with Mathew Syed’s commentary. Second part of the book is a little bit interesting and Third part of the book is a drag.

Overall, the message of the book is pretty clear , Deep Practice + Adequate & Appropriate Feedback + Growth Mindset are the factors behind the success of outstanding players.